Computerized tomography (CT) is a technique of reconstructing a cross-sectional image of an object from the radiographic projections of the object. FIG. 1 is a diagram schematically showing a typical X-ray CT apparatus.
In the typical CT apparatus, an X-ray source is moved around a target object, and irradiates X-rays to obtain the projections of the target object in many different directions. A cross-sectional image is obtained by subjecting the projections thus obtained to a computational operation, so-called reconstruction. The Filtered Back-Projection method (FBP) is commonly used to reconstruct a cross-sectional image from the projections. FBP is a kind of a transformation operation. In FBP, the projections are subjected to a filtering essentially equivalent to the differential filtering, followed by “back projection,”in which each projection is projected back along the original projection direction, thereby a cross-sectional image is obtained. In this case, the differential filtering usually amplifies noise or errors, which can be the source of artifacts (errors or false images which do not actually exist). Moreover, the back propagation operation spreads the artifacts thus created allover the cross-sectional image. Therefore, in CT, the artifacts often are not limited within a local portion around the source of the artifacts, and impairs the entire cross-sectional image, resulting in a fatal flaw.
Most artifacts are caused by the filtering operation and/or the back projection operation involved in FBP. Therefore, if FBP is not used, a cross-sectional image substantially can be free from most of artifacts. As a method of calculating a cross-sectional image other than FBP, the Algebraic Reconstruction Technique (ART) is historically important. ART was a major reconstruction method before FBP was proposed. In ART, the process of reconstruction is considered as a fitting problem where the cross-sectional image is a parameter and the projections are a target dataset to be fit. The cross-sectional image is iteratively modified so that projections (p) calculated from the cross-sectional image fit projections (p0) experimentally obtained. A feature of ART is that a cross-sectional image is asymptotically modified so that (p−p0) becomes zero. ART usually requires a vast computation time in comparison to FBP. Therefore, ART is currently used only for particular applications (the analysis of seismic waves, etc.). Although ART does not produce as extreme artifacts as FBP does, FBP often provides a more natural cross-sectional image than ART does.
Besides the filtering operation and the back projection operation, artifacts may be caused by a lack or shortage of data in projections. It is known that a lack or shortage of data often results in a fatal artifact especially in FBP. Other reconstruction techniques based on fitting, such as ART, are expected to be more robust against a lack or shortage of data than FBP. However, a lack of data is known to make CT an extremely “ill-posed problem,” under which it is essentially difficult to obtain reasonable solutions. One of the reasons why ART often fails in fitting is that ART uses (p−p0) for the target function of fitting. So it is quite natural to consider the use of the (p−p0)2 instead of (p−p0). In these cases, the least square method is one of the most popular way to minimize (p−p0)2. In the least square method, the inversion of a square matrix whose elements on one side is equal to the number of parameters is employed. Parameters in CT are values of pixels in the cross-sectional image, and therefore, the number of the parameters is huge, If a cross-sectional image has 1000×1000 pixels, the number of the parameters becomes a million, and the number of elements in the matrix is as huge as a trillion. Therefore, if the ordinary least square method is used, the matrix is too huge to calculate. Instead of the ordinary least square method, the Simultaneous Iterative Reconstruction Technique (SIRT) and the Iterative Least Square Technique (ILST) have been proposed. In these techniques, the calculation of a cross-sectional image is considered as a fitting problem as in ART. FBP is utilized as an inverse operation to calculate the cross-sectional image from projections partway through the calculation in both SIRT and WLST so as to circumvent the use of the ordinary least square method as described above. Therefore, none of SIRT and ILST does not substantially solve the problems involved with the filtering operation and the back projection operation as in FBP. This is probably why there have been reports that SIRT and ILST just “reduce” artifacts.    Non-Patent Document 1: Yazdi M., Gingras L., Beaulieu L., An adaptive approach to metal artifact reduction in helical computed tomography for radiation therapy treatment planning: experimental and clinical studies, Int J Radiat Oncol Biol Phys, 62: 1224-1231, 2005.